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<div class="title">FemBases.py</div>  </div>
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<div class="fragment"><pre class="fragment"><a name="l00001"></a>00001 <span class="comment">#!/usr/bin/python</span>
<a name="l00002"></a>00002 <span class="comment"># -*- coding: utf-8 -*-</span>
<a name="l00003"></a>00003 
<a name="l00004"></a>00004 <span class="comment"># Copyright (c) 2011</span>
<a name="l00005"></a>00005 <span class="comment">#</span>
<a name="l00006"></a>00006 <span class="comment"># Permission is hereby granted, free of charge, to any person obtaining a</span>
<a name="l00007"></a>00007 <span class="comment"># copy of this software and associated documentation files (the &quot;Software&quot;),</span>
<a name="l00008"></a>00008 <span class="comment"># to deal in the Software without restriction, including without limitation</span>
<a name="l00009"></a>00009 <span class="comment"># the rights to use, copy, modify, merge, publish, distribute, sublicense,</span>
<a name="l00010"></a>00010 <span class="comment"># and/or sell copies of the Software, and to permit persons to whom the</span>
<a name="l00011"></a>00011 <span class="comment"># Software is furnished to do so, subject to the following conditions:</span>
<a name="l00012"></a>00012 <span class="comment">#</span>
<a name="l00013"></a>00013 <span class="comment"># The above copyright notice and this permission notice shall be included in</span>
<a name="l00014"></a>00014 <span class="comment"># all copies or substantial portions of the Software.</span>
<a name="l00015"></a>00015 <span class="comment">#</span>
<a name="l00016"></a>00016 <span class="comment"># THE SOFTWARE IS PROVIDED &quot;AS IS&quot;, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR</span>
<a name="l00017"></a>00017 <span class="comment"># IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,</span>
<a name="l00018"></a>00018 <span class="comment"># FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE</span>
<a name="l00019"></a>00019 <span class="comment"># AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER</span>
<a name="l00020"></a>00020 <span class="comment"># LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,</span>
<a name="l00021"></a>00021 <span class="comment"># OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE</span>
<a name="l00022"></a>00022 <span class="comment"># SOFTWARE.</span>
<a name="l00023"></a>00023 <span class="comment">#</span>
<a name="l00024"></a>00024 <span class="comment"># Author: Jesus Carrero &lt;j.o.carrero@gmail.com&gt;</span>
<a name="l00025"></a>00025 <span class="comment"># Mountain View, CA</span>
<a name="l00026"></a>00026 
<a name="l00027"></a>00027 <span class="keyword">from</span> scipy <span class="keyword">import</span> r_, array, zeros, ones
<a name="l00028"></a>00028 
<a name="l00029"></a>00029 <span class="keyword">class </span>FemBasisBase(object):
<a name="l00030"></a>00030 
<a name="l00031"></a>00031     __slots__ = [<span class="stringliteral">&#39;m_b_type&#39;</span>, <span class="stringliteral">&#39;m_mass&#39;</span>, <span class="stringliteral">&#39;m_deg_of_freedom&#39;</span>, <span class="stringliteral">&#39;m_support&#39;</span>]
<a name="l00032"></a>00032 
<a name="l00033"></a>00033     <span class="keyword">def </span><a class="code" href="classsolvers_1_1EuropeanTimeStepper_1_1EuropeanTimeStepper.html#ac6e50f8c1345b947f2bdf5b47d3524b2">__init__</a>(self, b_type, _dof, support):
<a name="l00034"></a>00034         <span class="stringliteral">&quot;&quot;&quot;   &quot;&quot;&quot;</span>
<a name="l00035"></a>00035 
<a name="l00036"></a>00036         self.m_b_type = b_type
<a name="l00037"></a>00037         self.m_deg_of_freedom = _dof
<a name="l00038"></a>00038         self.m_support = support
<a name="l00039"></a>00039         self.m_mass = <span class="keywordtype">None</span>
<a name="l00040"></a>00040         self._mass()
<a name="l00041"></a>00041 
<a name="l00042"></a>00042     <span class="keyword">def </span>_mass(self):
<a name="l00043"></a>00043         <span class="stringliteral">&quot;&quot;&quot; virtual method to call mass matrix.&quot;&quot;&quot;</span>
<a name="l00044"></a>00044 
<a name="l00045"></a>00045         <span class="keyword">assert</span> <span class="keyword">False</span>
<a name="l00046"></a>00046 
<a name="l00047"></a>00047     <span class="keyword">def </span>dof(self):
<a name="l00048"></a>00048         <span class="stringliteral">&quot;&quot;&quot; degrees of freedom   &quot;&quot;&quot;</span>
<a name="l00049"></a>00049 
<a name="l00050"></a>00050         <span class="keywordflow">return</span> self.m_deg_of_freedom
<a name="l00051"></a>00051 
<a name="l00052"></a>00052     <span class="keyword">def </span>shape_func_support(self):
<a name="l00053"></a>00053         <span class="stringliteral">&quot;&quot;&quot; return extrems defineing domain for shape functions. &quot;&quot;&quot;</span>
<a name="l00054"></a>00054 
<a name="l00055"></a>00055         <span class="keywordflow">return</span> self.m_support
<a name="l00056"></a>00056 
<a name="l00057"></a>00057     <span class="keyword">def </span>get_mass_mat(self):
<a name="l00058"></a>00058         <span class="stringliteral">&quot;&quot;&quot; get mass matrix.  &quot;&quot;&quot;</span>
<a name="l00059"></a>00059 
<a name="l00060"></a>00060         <span class="keywordflow">return</span> self.m_mass
<a name="l00061"></a>00061 
<a name="l00062"></a>00062     <span class="keyword">def </span>integrate(self, i, j):
<a name="l00063"></a>00063         <span class="stringliteral">&quot;&quot;&quot; integrate two shape functions in reference interval.&quot;&quot;&quot;</span>
<a name="l00064"></a>00064 
<a name="l00065"></a>00065         <span class="keywordflow">return</span> self.m_mass[i, j]
<a name="l00066"></a>00066 
<a name="l00067"></a>00067 
<a name="l00068"></a><a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html">00068</a> <span class="keyword">class </span><a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html">LagrangeP1</a>(FemBasisBase):
<a name="l00069"></a>00069 
<a name="l00070"></a>00070     <span class="stringliteral">&quot;&quot;&quot; compute the inner product</span>
<a name="l00071"></a>00071 <span class="stringliteral">      ------- xi \in [0, 1]</span>
<a name="l00072"></a>00072 <span class="stringliteral">  &quot;&quot;&quot;</span>
<a name="l00073"></a>00073 
<a name="l00074"></a><a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#ae14835cb69ead099763600ebcc5d1f7e">00074</a>     <span class="keyword">def </span><a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#ae14835cb69ead099763600ebcc5d1f7e">__init__</a>(self):
<a name="l00075"></a>00075         <span class="stringliteral">&quot;&quot;&quot; Instantiate an obj of type P1.  &quot;&quot;&quot;</span>
<a name="l00076"></a>00076 
<a name="l00077"></a>00077         FemBasisBase.__init__(self, <span class="stringliteral">&#39;lagrangeP1&#39;</span>, 2, array([0, 1]))
<a name="l00078"></a>00078 
<a name="l00079"></a><a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#a73fab949bab4f30e95d7452e04b9314d">00079</a>     <span class="keyword">def </span><a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#a73fab949bab4f30e95d7452e04b9314d">l1</a>(self, points):
<a name="l00080"></a>00080         <span class="stringliteral">&quot;&quot;&quot;  L_1(pointsi) = 1.0-pointsi &quot;&quot;&quot;</span>
<a name="l00081"></a>00081 
<a name="l00082"></a>00082         <span class="keywordflow">return</span> 1. - points.reshape(points.size, 1)
<a name="l00083"></a>00083 
<a name="l00084"></a><a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#a42d1cfdc656dfef1e059a791c6672789">00084</a>     <span class="keyword">def </span><a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#a42d1cfdc656dfef1e059a791c6672789">l2</a>(self, points):
<a name="l00085"></a>00085         <span class="stringliteral">&quot;&quot;&quot; L_2(pointsi) = pointsi &quot;&quot;&quot;</span>
<a name="l00086"></a>00086 
<a name="l00087"></a>00087         <span class="keywordflow">return</span> points.reshape(points.size, 1)
<a name="l00088"></a>00088 
<a name="l00089"></a>00089     <span class="keyword">def </span>_mass(self):
<a name="l00090"></a>00090         <span class="stringliteral">&quot;&quot;&quot; compute mass matrix. &quot;&quot;&quot;</span>
<a name="l00091"></a>00091 
<a name="l00092"></a>00092         self.<a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#a5780d3965c0231010649a9800398800c">m_mass</a> = zeros((2, 2), <span class="stringliteral">&#39;float64&#39;</span>)
<a name="l00093"></a>00093         self.<a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#a5780d3965c0231010649a9800398800c">m_mass</a>[0, :] = array([1., 1 / 2.])
<a name="l00094"></a>00094         self.<a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#a5780d3965c0231010649a9800398800c">m_mass</a>[1, :] = array([1 / 2., 1.])
<a name="l00095"></a>00095         self.<a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#a5780d3965c0231010649a9800398800c">m_mass</a> /= 3.
<a name="l00096"></a>00096 
<a name="l00097"></a>00097     @classmethod
<a name="l00098"></a><a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#aaa3b425ea5e1103d00b452392f0b5df6">00098</a>     <span class="keyword">def </span><a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#aaa3b425ea5e1103d00b452392f0b5df6">get_grad_grad</a>(cls):
<a name="l00099"></a>00099         <span class="stringliteral">&quot;&quot;&quot; return grad-grad matrix.  &quot;&quot;&quot;</span>
<a name="l00100"></a>00100 
<a name="l00101"></a>00101         grgr = ones((2, 2), <span class="stringliteral">&#39;float&#39;</span>)
<a name="l00102"></a>00102         grgr[0, :] = array([1., -1.])
<a name="l00103"></a>00103         grgr[1, :] = array([-1., 1.])
<a name="l00104"></a>00104         <span class="keywordflow">return</span> grgr
<a name="l00105"></a>00105 
<a name="l00106"></a>00106 
<a name="l00107"></a>00107     @classmethod
<a name="l00108"></a><a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#a8af91f22a469068e376d76e061e37f2f">00108</a>     <span class="keyword">def </span><a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#a8af91f22a469068e376d76e061e37f2f">get_grad_disp</a>(cls):
<a name="l00109"></a>00109         <span class="stringliteral">&quot;&quot;&quot; returns grad-disp matrix.  &quot;&quot;&quot;</span>
<a name="l00110"></a>00110 
<a name="l00111"></a>00111         grds = ones((2, 2), <span class="stringliteral">&#39;float&#39;</span>)
<a name="l00112"></a>00112         grds[0, :] = array([-1., 1.])
<a name="l00113"></a>00113         grds[1, :] = array([-1., 1.])
<a name="l00114"></a>00114         <span class="keywordflow">return</span> grds / 2.
<a name="l00115"></a>00115 
<a name="l00116"></a><a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#aa2963401569410f667188212f5695723">00116</a>     <span class="keyword">def </span><a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#aa2963401569410f667188212f5695723">values</a>(self, points):
<a name="l00117"></a>00117         <span class="stringliteral">&quot;&quot;&quot; evalue shape function at quad points.</span>
<a name="l00118"></a>00118 <span class="stringliteral">            quad formulas are defined in the interval [-1, 1].</span>
<a name="l00119"></a>00119 <span class="stringliteral">        &quot;&quot;&quot;</span>
<a name="l00120"></a>00120         qpoints1 = (points + 1.) / 2.
<a name="l00121"></a>00121         basis_at_points = [self.<a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#a73fab949bab4f30e95d7452e04b9314d">l1</a>(qpoints1)]
<a name="l00122"></a>00122         basis_at_points.append(self.<a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#a42d1cfdc656dfef1e059a791c6672789">l2</a>(qpoints1))
<a name="l00123"></a>00123         basis_at_points = array(basis_at_points)
<a name="l00124"></a>00124         basis_at_points = basis_at_points.reshape((2, points.size))
<a name="l00125"></a>00125         <span class="keywordflow">return</span> basis_at_points
<a name="l00126"></a>00126 
<a name="l00127"></a>00127     @classmethod
<a name="l00128"></a><a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#a16e2064331c1575f2698f82054641434">00128</a>     <span class="keyword">def </span><a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#a16e2064331c1575f2698f82054641434">derivatives</a>(cls, points):
<a name="l00129"></a>00129         <span class="stringliteral">&quot;&quot;&quot; evaluate derivatives at quad points.&quot;&quot;&quot;</span>
<a name="l00130"></a>00130 
<a name="l00131"></a>00131         <span class="keywordflow">return</span> r_[-ones((1, points.size)), ones((1, points.size))]
<a name="l00132"></a>00132 
<a name="l00133"></a>00133     @classmethod
<a name="l00134"></a><a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#af8104a2bd4e31297a8eb3e68711ee015">00134</a>     <span class="keyword">def </span><a class="code" href="classutils_1_1FemBases_1_1LagrangeP1.html#af8104a2bd4e31297a8eb3e68711ee015">second_der</a>(cls, points):
<a name="l00135"></a>00135         <span class="stringliteral">&quot;&quot;&quot; second derivative at quad points.&quot;&quot;&quot;</span>
<a name="l00136"></a>00136 
<a name="l00137"></a>00137         <span class="keywordflow">return</span> zeros((2, points.size))
<a name="l00138"></a>00138 
<a name="l00139"></a><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html">00139</a> <span class="keyword">class </span><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html">LagrangeP2</a>(FemBasisBase):
<a name="l00140"></a>00140 
<a name="l00141"></a>00141     <span class="stringliteral">&quot;&quot;&quot; compute the inner product</span>
<a name="l00142"></a>00142 <span class="stringliteral">      ------- xi \in [-1, 1]</span>
<a name="l00143"></a>00143 <span class="stringliteral">  &quot;&quot;&quot;</span>
<a name="l00144"></a>00144 
<a name="l00145"></a><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#a17fbfeb603830e651f7596739459ba45">00145</a>     <span class="keyword">def </span><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#a17fbfeb603830e651f7596739459ba45">__init__</a>(self):
<a name="l00146"></a>00146         <span class="stringliteral">&quot;&quot;&quot; Instantiate an object to P2 type. &quot;&quot;&quot;</span>
<a name="l00147"></a>00147 
<a name="l00148"></a>00148         FemBasisBase.__init__(self, <span class="stringliteral">&#39;LagrangeP2&#39;</span>, 3, array([-1, 1]))
<a name="l00149"></a>00149 
<a name="l00150"></a>00150     @classmethod
<a name="l00151"></a><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#adc47860be69ef0d4dd395b1253deb93e">00151</a>     <span class="keyword">def </span><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#adc47860be69ef0d4dd395b1253deb93e">l1</a>(cls, points):
<a name="l00152"></a>00152         <span class="stringliteral">&quot;&quot;&quot; L_1(pointsi) = -0.5*pointsi*(1.0-pointsi) &quot;&quot;&quot;</span>
<a name="l00153"></a>00153 
<a name="l00154"></a>00154         val = -.5 * points * (1. - points)
<a name="l00155"></a>00155         <span class="keywordflow">return</span> val.reshape(val.size, 1)
<a name="l00156"></a>00156 
<a name="l00157"></a>00157     @classmethod
<a name="l00158"></a><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#a57100c2a57f2edc0583ded9f372ff978">00158</a>     <span class="keyword">def </span><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#a57100c2a57f2edc0583ded9f372ff978">l2</a>(cls, points):
<a name="l00159"></a>00159         <span class="stringliteral">&quot;&quot;&quot;  N_2(pointsi) = 1.0-pointsi**2 &quot;&quot;&quot;</span>
<a name="l00160"></a>00160 
<a name="l00161"></a>00161         val = 1. - points ** 2
<a name="l00162"></a>00162         <span class="keywordflow">return</span> val.reshape(val.size, 1)
<a name="l00163"></a>00163 
<a name="l00164"></a>00164     @classmethod
<a name="l00165"></a><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#af64d5f5e7dd42e9b802305ce9c6eb3a3">00165</a>     <span class="keyword">def </span><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#af64d5f5e7dd42e9b802305ce9c6eb3a3">l3</a>(cls, points):
<a name="l00166"></a>00166         <span class="stringliteral">&quot;&quot;&quot; N_3(pointsi) =  0.5*pointsi*(1+pointsi) &quot;&quot;&quot;</span>
<a name="l00167"></a>00167 
<a name="l00168"></a>00168         val = .5 * points * (1 + points)
<a name="l00169"></a>00169         <span class="keywordflow">return</span> val.reshape(val.size, 1)
<a name="l00170"></a>00170 
<a name="l00171"></a>00171     <span class="keyword">def </span>_mass(self):
<a name="l00172"></a>00172         <span class="stringliteral">&quot;&quot;&quot; compute mass matrix.  &quot;&quot;&quot;</span>
<a name="l00173"></a>00173 
<a name="l00174"></a>00174         self.<a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#aa3e3de66b0f616f4a1064c34e972e3fb">m_mass</a> = zeros((3, 3), <span class="stringliteral">&#39;float64&#39;</span>)
<a name="l00175"></a>00175         self.<a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#aa3e3de66b0f616f4a1064c34e972e3fb">m_mass</a>[0, :] = array([4., 2., -1.])
<a name="l00176"></a>00176         self.<a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#aa3e3de66b0f616f4a1064c34e972e3fb">m_mass</a>[1, :] = array([2., 16., 2.])
<a name="l00177"></a>00177         self.<a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#aa3e3de66b0f616f4a1064c34e972e3fb">m_mass</a>[2, :] = array([-1., 2., 4.])
<a name="l00178"></a>00178         self.<a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#aa3e3de66b0f616f4a1064c34e972e3fb">m_mass</a> /= 30.
<a name="l00179"></a>00179 
<a name="l00180"></a>00180     @classmethod
<a name="l00181"></a><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#ac51d628dcf4f7d5d746ad5242c9142b0">00181</a>     <span class="keyword">def </span><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#ac51d628dcf4f7d5d746ad5242c9142b0">get_grad_grad</a>(cls):
<a name="l00182"></a>00182         <span class="stringliteral">&quot;&quot;&quot; compute grad-grad at reference element.  &quot;&quot;&quot;</span>
<a name="l00183"></a>00183 
<a name="l00184"></a>00184         grgr = ones((3, 3), <span class="stringliteral">&#39;float&#39;</span>)
<a name="l00185"></a>00185         grgr[0, :] = array([7., -8., 1.])
<a name="l00186"></a>00186         grgr[1, :] = array([-8., 16., -8.])
<a name="l00187"></a>00187         grgr[2, :] = array([1., -8., 7.])
<a name="l00188"></a>00188         <span class="keywordflow">return</span> grgr / 3.
<a name="l00189"></a>00189 
<a name="l00190"></a>00190     @classmethod
<a name="l00191"></a><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#a5b9cf434edd8071a8c35a8e8cb97e74e">00191</a>     <span class="keyword">def </span><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#a5b9cf434edd8071a8c35a8e8cb97e74e">get_grad_disp</a>(cls):
<a name="l00192"></a>00192         <span class="stringliteral">&quot;&quot;&quot; compute matrix grad-displacment.  &quot;&quot;&quot;</span>
<a name="l00193"></a>00193 
<a name="l00194"></a>00194         grds = ones((3, 3), <span class="stringliteral">&#39;float&#39;</span>)
<a name="l00195"></a>00195         int_ni = array([1., 4., 1.]) / 3.
<a name="l00196"></a>00196         intx_ni = array([-1., 0., 1.]) / 3.
<a name="l00197"></a>00197 
<a name="l00198"></a>00198         grds[0, :] = -.5 * (int_ni - 2 * intx_ni)
<a name="l00199"></a>00199         grds[1, :] = -2. * intx_ni
<a name="l00200"></a>00200         grds[2, :] = .5 * (int_ni + 2 * intx_ni)
<a name="l00201"></a>00201         <span class="keywordflow">return</span> grds.T
<a name="l00202"></a>00202 
<a name="l00203"></a><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#a0d03bbf9250b21e044f77e9597d9f6a1">00203</a>     <span class="keyword">def </span><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#a0d03bbf9250b21e044f77e9597d9f6a1">values</a>(self, points):
<a name="l00204"></a>00204         <span class="stringliteral">&quot;&quot;&quot; evaluate shape function at points.&quot;&quot;&quot;</span>
<a name="l00205"></a>00205 
<a name="l00206"></a>00206         basis_at_points = zeros((3, points.size))
<a name="l00207"></a>00207         basis_at_points[0, : ] =  self.<a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#adc47860be69ef0d4dd395b1253deb93e">l1</a>(points).flatten()
<a name="l00208"></a>00208         basis_at_points[1, : ] =  self.<a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#a57100c2a57f2edc0583ded9f372ff978">l2</a>(points).flatten()
<a name="l00209"></a>00209         basis_at_points[2, : ] =  self.<a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#af64d5f5e7dd42e9b802305ce9c6eb3a3">l3</a>(points).flatten()
<a name="l00210"></a>00210         <span class="keywordflow">return</span> basis_at_points
<a name="l00211"></a>00211 
<a name="l00212"></a>00212     @classmethod
<a name="l00213"></a><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#aa61c8d7ac412cb06175920f58e5518d8">00213</a>     <span class="keyword">def </span><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#aa61c8d7ac412cb06175920f58e5518d8">derivatives</a>(cls, points):
<a name="l00214"></a>00214         <span class="stringliteral">&quot;&quot;&quot; evaluate at the derivatives of the shape fucntions. &quot;&quot;&quot;</span>
<a name="l00215"></a>00215 
<a name="l00216"></a>00216         tmp = points.reshape((1, points.size))
<a name="l00217"></a>00217         <span class="keywordflow">return</span> r_[r_[tmp - .5, -2. * tmp], tmp + .5]
<a name="l00218"></a>00218 
<a name="l00219"></a>00219     @classmethod
<a name="l00220"></a><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#a7b781236f4fd0b13ee3129ac890774f8">00220</a>     <span class="keyword">def </span><a class="code" href="classutils_1_1FemBases_1_1LagrangeP2.html#a7b781236f4fd0b13ee3129ac890774f8">second_der</a>(cls, points):
<a name="l00221"></a>00221         <span class="stringliteral">&quot;&quot;&quot; second derivative of shape functions. &quot;&quot;&quot;</span>
<a name="l00222"></a>00222 
<a name="l00223"></a>00223         tmp = ones( (3, points.size) )
<a name="l00224"></a>00224         tmp[1, :] *= -2.
<a name="l00225"></a>00225         <span class="keywordflow">return</span> tmp
<a name="l00226"></a>00226 
<a name="l00227"></a>00227 
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